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MgNO: Efficient Parameterization of Linear Operators via Multigrid
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Proposes MgNO, a neural operator that treats neuron connections as bounded linear operators parameterized by multi-channel multigrid V-cycles (convolutions), removing lifting/projection layers and handling boundary conditions naturally.
Problem
Learning solution operators of PDEs efficiently and accurately, with existing neural operators (spectral, wavelet, CNN-based) facing limits in parameter efficiency, boundary-condition handling, overfitting, and accuracy.
Approach
Generalizes fully connected neural networks to neural operators whose inter-neuron connections are bounded linear operators, and parameterizes those operators with convolutional multigrid (V-cycle) structures, which eliminates lifting/projection layers and naturally encodes Dirichlet, Neumann, or periodic boundary conditions.
Key results
- On Darcy rough, MgNO reaches relative L2 error 0.339e-2 with 0.57M parameters, versus 1.023e-2 for U-NO (16.39M) and 1.613e-2 for FNO2D (2.37M).
- On Navier-Stokes NS2, MgNO achieves relative error 1.63e-2 vs 2.64e-2 for U-NO and 3.09e-2 for MWT.
- On Helmholtz, MgNO achieves L2 error 0.71e-2 vs 1.26e-2 for U-NO, with 7.58M parameters.
Measured effects reported in the paper
| What was changed | Where | Measured | Effect |
|---|---|---|---|
| Use MgNO instead of U-NO | operator learning model · Darcy smooth | relative L² error (×10⁻²): 0.492 -> 0.176 | Helped |
| Use MgNO instead of U-NO | operator learning model · Darcy rough | relative L² error (×10⁻²): 1.023 -> 0.339 | Helped |
| Use MgNO instead of DilResNet | operator learning model · Darcy multiscale | relative H¹ error (×10⁻²): 3.528 -> 1.756 | Helped |
| Train using the NS2 configuration rather than the original Navier-Stokes setup | MgNO · Navier Stokes, ν=1e-5 | error: 10.82 -> 1.63 | Helped |
| Train on resolution 64 and evaluate at higher resolution using MgNO instead of FNO | operator learning model · Darcy multiscale, test resolution 512 | relative L² error (×10⁻²): 9.1054 -> 1.205 | Helped |
| Use 4 levels instead of 5 (ablation) | MgNO · Navier Stokes NS2 | L² error (×10⁻²): 1.63 -> 2.10 | Hurt |
| Use 3 levels instead of 5 (ablation) | MgNO · Navier Stokes NS2 | L² error (×10⁻²): 1.63 -> 2.44 | Hurt |
| Add boundary-condition handling (ablation) | MgNO · Navier Stokes NS2 | L² error (×10⁻²): 3.18 -> 1.63 | Helped |
| Increase depth from 5 to 6 layers (ablation) | MgNO · Navier Stokes NS2 | L² error (×10⁻²): 1.63 -> 1.47 | Helped |
| Replace multigrid parameterization with spectral convolution (ablation) | MgNO · Navier Stokes NS2 | L² error (×10⁻²): 1.63 -> 3.91 | Hurt |
Surprising findings
- MgNO-high-in using 512x512 inputs reduces Darcy rough L2 error to 0.275e-2, showing high-resolution inputs help fine-scale predictions.
- The paper reports MgNO trains without overfitting while larger baselines (MWT, UNO, LSM) reach comparable training error but much higher test error.
Limitations
- Irregular domains require a deformation mapping to a regular grid; not benchmarked natively on general point clouds.
- Performance is sensitive to multigrid depth, and deeper configurations increase computational cost.
- Theoretical convergence constant is only quantified numerically on one example.
Experimental setup
Datasets: Darcy smooth Darcy rough Darcy multiscale Navier-Stokes (nu=1e-5) Helmholtz Pipe
Benchmarks: Darcy smooth Darcy rough Darcy multiscale Navier-Stokes Pipe Helmholtz
Metrics: relative L2 error relative H1 error parameter count runtime per iteration
Baselines: FNO UNet U-NO MWT Galerkin transformer LSM DilResNet sFNO-v2
Builds on: MgNet Fourier neural operator U-Net Galerkin transformer
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